We use spectral proper orthogonal decomposition (SPOD) to extract and analyse coherent structures in the turbulent wake of a disk at Reynolds number
$ {\textit {Re}} = 5 \times 10^{4}$
and Froude numbers
$ {\textit {Fr}} = 2$
, 10. We find that the SPOD eigenspectra of both wakes exhibit a low-rank behaviour and the relative contribution of low-rank modes to total fluctuation energy increases with
$x/D$
. The vortex shedding (VS) mechanism, which corresponds to
$ {\textit {St}} \approx 0.11 - 0.13$
in both wakes, is active and dominant throughout the domain in both wakes. The continual downstream decay of the SPOD eigenspectrum peak at the VS mode, which is a prominent feature of the unstratified wake, is inhibited by buoyancy, particularly for
$ {\textit {Fr}} = 2$
. The energy at and near the VS frequency is found to appear in the outer region of the wake when the downstream distance exceeds
$Nt = Nx/U = 6 - 8$
. Visualizations show that unsteady internal gravity waves (IGWs) emerge at the same
$Nt = 6 - 8$
. A causal link between the VS mechanism and the unsteady IGW generation is also established using the SPOD-based reconstruction and analysis of the pressure transport term. These IGWs are also picked up in SPOD analysis as a structural change in the shape of the leading SPOD eigenmode. The
$ {\textit {Fr}} = 2$
wake shows layering in the wake core at
$Nt > 15$
which is captured by the leading SPOD eigenmodes of the VS frequency at downstream locations
$x/D > 30$
. The VS mode of the
$ {\textit {Fr}} = 2$
wake is streamwise coherent, consisting of
$V$
-shaped structures at
$x/D \gtrsim 30$
. Overall, we find that the coherence of wakes, initiated by the VS mode at the body, is prolonged by buoyancy to far downstream. Also, this coherence is spatially modified by buoyancy into horizontal layers and IGWs. Low-order truncations of SPOD modes are shown to efficiently reconstruct important second-order statistics.
We use spectral proper orthogonal decomposition (SPOD) to extract and analyse coherent structures in the turbulent wake of a disk at Reynolds number
$ {\textit {Re}} = 5 \times 10^{4}$
and Froude numbers
$ {\textit {Fr}} = 2$
, 10. We find that the SPOD eigenspectra of both wakes exhibit a low-rank behaviour and the relative contribution of low-rank modes to total fluctuation energy increases with
$x/D$
. The vortex shedding (VS) mechanism, which corresponds to
$ {\textit {St}} \approx 0.11 - 0.13$
in both wakes, is active and dominant throughout the domain in both wakes. The continual downstream decay of the SPOD eigenspectrum peak at the VS mode, which is a prominent feature of the unstratified wake, is inhibited by buoyancy, particularly for
$ {\textit {Fr}} = 2$
. The energy at and near the VS frequency is found to appear in the outer region of the wake when the downstream distance exceeds
$Nt = Nx/U = 6 - 8$
. Visualizations show that unsteady internal gravity waves (IGWs) emerge at the same
$Nt = 6 - 8$
. A causal link between the VS mechanism and the unsteady IGW generation is also established using the SPOD-based reconstruction and analysis of the pressure transport term. These IGWs are also picked up in SPOD analysis as a structural change in the shape of the leading SPOD eigenmode. The
$ {\textit {Fr}} = 2$
wake shows layering in the wake core at
$Nt > 15$
which is captured by the leading SPOD eigenmodes of the VS frequency at downstream locations
$x/D > 30$
. The VS mode of the
$ {\textit {Fr}} = 2$
wake is streamwise coherent, consisting of
$V$
-shaped structures at
$x/D \gtrsim 30$
. Overall, we find that the coherence of wakes, initiated by the VS mode at the body, is prolonged by buoyancy to far downstream. Also, this coherence is spatially modified by buoyancy into horizontal layers and IGWs. Low-order truncations of SPOD modes are shown to efficiently reconstruct important second-order statistics.
“…The frequency range in this comparison is extended to to show the model performance beyond the range for which it has been tuned. We limit the analysis to this range since, below , the hydrodynamic fluctuations reach the acoustic field as reported by Nekkanti & Schmidt (2021), who used the same database for the cases M09Mc00 and M07Mc00; and at , the acoustic level is already 20 dB less than the peak in all the cases. Figure 21.PSD of the acoustic pressure obtained from LES (solid) and predicted by the line-source model (dashed) for ( a , b ) static jet cases and ( c , d ) cases with flight stream at two different propagation angles, ( a , c ) and ( b , d ) .…”
Section: Empirical Modelling Of the Acoustically Efficient Forcing Co...mentioning
Modelling the noise emitted by turbulent jets is made difficult by their acoustic inefficiency: only a tiny fraction of the near-field turbulent kinetic energy is propagated to the far field as acoustic waves. As a result, jet-noise models must accurately capture this small, acoustically efficient component hidden among comparatively inefficient fluctuations. In this paper, we identify this acoustically efficient near-field source from large-eddy simulation data and use it to inform a predictive model. Our approach uses the resolvent framework, in which the source takes the form of nonlinear fluctuation terms that act as a forcing on the linearised Navier–Stokes equations. First, we identify the forcing that, when acted on by the resolvent operator, produces the leading spectral proper orthogonal decomposition modes in the acoustic field for a Mach 0.4 jet. Second, the radiating components of this forcing are isolated by retaining only portions with a supersonic phase speed. This component makes up less than 0.05 % of the total forcing energy but generates most of the acoustic response, especially at peak (downstream) radiation angles. Finally, we propose an empirical model for the identified acoustically efficient forcing components. The model is tested at other Mach numbers and flight-stream conditions and predicts noise within 2 dB accuracy for a range of frequencies, downstream angles and flight conditions.
“…2018; Pickering et al. 2019; Nekkanti & Schmidt 2021), the wake behind a disk (Johansson, George & Woodward 2002; Johansson & George 2006 a , b ; Tutkun, Johansson & George 2008; Ghate, Towne & Lele 2020; Nidhan et al. 2020), pipe flows (Hellström & Smits 2014; Hellström, Ganapathisubramani & Smits 2015; Hellström, Marusic & Smits 2016; Hellström & Smits 2017) and channel flows (Muralidhar et al.…”
Four different applications of spectral proper orthogonal decomposition (SPOD) are demonstrated on large-eddy simulation data of a turbulent jet. These are: low-rank reconstruction, denoising, frequency–time analysis and prewhitening. We demonstrate SPOD-based flow-field reconstruction using direct inversion of the SPOD algorithm (frequency-domain approach) and propose an alternative approach based on projection of the time series data onto the modes (time-domain approach). We further present a SPOD-based denoising strategy that is based on hard thresholding of the SPOD eigenvalues. The proposed strategy achieves significant noise reduction while facilitating drastic data compression. In contrast to standard methods of frequency–time analysis such as wavelet transform, a proposed SPOD-based approach yields a spectrogram that characterises the temporal evolution of spatially coherent flow structures. A convolution-based strategy is proposed to compute the time-continuous expansion coefficients. When applied to the turbulent jet data, SPOD-based frequency–time analysis reveals that the intermittent occurrence of large-scale coherent structures is directly associated with high-energy events. This work suggests that the time-domain approach is preferable for low-rank reconstruction of individual snapshots, and the frequency-domain approach for denoising and frequency–time analysis.
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